Sparse halves in $K_4$-free graphs

TL;DR

This paper investigates a conjecture about $K_4$-free graphs, proving it for the special case of regular graphs, thus advancing understanding of their structural properties.

Contribution

The paper proves the Chung-Graham conjecture for all regular graphs, providing a significant partial validation of the conjecture.

Findings

The conjecture holds for regular $K_4$-free graphs.

Regular graphs satisfy the conjecture's conditions.

Progress towards the general conjecture for all $K_4$-free graphs.

Abstract

A conjecture of Chung and Graham states that every $K_4$-free graph on $n$ vertices contains a vertex set of size $\lfloor n/2 \rfloor$ that spans at most $n^2/18$ edges. We make the first step toward this conjecture by showing that it holds for all regular graphs.